This is similar to our observation that the state space curvature becomes negative in the immediate vicinity of the quantum critical point, but our construction pertains to a state space manifold that might be applied to transitions where the magnetic field is not a priori the driving parameter of the phase transition.
It is also interesting that there is a discontinuous jump from a region of positive state space curvature in the Haldane phase and immediately around the critical points to a region of negative state space curvature in the quantum critical regime, without any intermediate flat region. In the grey regions of Fig. REF , it is important to emphasize that the form of the quantum metric defined in Eq. REF  breaks down, as it does not account for the non-adiabatic level crossings that occur at the critical point. A more careful treatment would involve computing the entire spectrum and applying the form of the quantum metric introduced in [1]}. We leave these questions to future study.
